Tn 110 lecture 2 logic
- 1. LAWS OF ALGEBRA OF PREPOSITIONS Mayengo, M. M 1 Identities 𝑝 ∨ 𝑝 ≡ 𝑝 𝑝 ∨ 𝑇 ≡ 𝑇 𝑝 ∨ 𝐹 ≡ 𝑝 𝑝 ∧ 𝑝 ≡ 𝑝 𝑝 ∧ 𝑇 ≡ 𝑝 𝑝 ∧ 𝐹 ≡ 𝐹 𝑝 → 𝑝 ≡ 𝑇 𝑝 → 𝑇 ≡ 𝑇 𝑝 → 𝐹 ≡∼ 𝑝 𝑇 → 𝑝 ≡ 𝑝 𝐹 → 𝑝 ≡ 𝑇 𝑝 ↔ 𝑝 ≡ 𝑇 𝑝 ↔ 𝑇 ≡ 𝑝 𝑝 ↔ 𝐹 ≡∼ 𝑝
- 2. LAWS OF ALGEBRA OF PREPOSITIONS Mayengo, M. M 2 Commutative Law 𝑝 ∨ 𝑞 ≡ 𝑞 ∨ 𝑝 𝑝 ∧ 𝑞 ≡ 𝑞 ∧ 𝑝 𝑝 → 𝑞 ≠ 𝑞 → 𝑝 𝑝 ↔ 𝑞 ≡ 𝑞 ↔ 𝑝 The Law of Double Negation ∼ ∼ 𝑝 ≡ 𝑝 Complement Law 𝑝 ∨∼ 𝑝 ≡ 𝑇 𝑝 ∧∼ 𝑝 ≡ 𝐹 𝑝 →∼ 𝑝 ≡∼ 𝑝 ∼ 𝑝 → 𝑝 ≡ 𝑝 𝑝 ↔∼ 𝑝 ≡ 𝐹
- 3. LAWS OF ALGEBRA OF PREPOSITIONS Mayengo, M. M 3 Associative Law 𝑝 ∨ 𝑞 ∨ 𝑟 ≡ 𝑝 ∨ 𝑞 ∨ 𝑟 𝑝 ∧ 𝑞 ∧ 𝑟 ≡ 𝑝 ∧ 𝑞 ∧ 𝑟 Distributive Law 𝑝 ∨ 𝑞 ∧ 𝑟 ≡ 𝑝 ∨ 𝑞 ∧ (𝑝 ∨ 𝑟) 𝑝 ∧ 𝑞 ∨ 𝑟 ≡ 𝑝 ∧ 𝑞 ∨ (𝑝 ∧ 𝑟)
- 4. LAWS OF ALGEBRA OF PREPOSITIONS Mayengo, M. M 4 The Law of Absorption 𝑝 ∨ 𝑝 ∧ 𝑞 ≡ 𝑝 𝑝 ∧ 𝑝 ∨ 𝑞 ≡ 𝑝 De Morgan’s Law ∼ 𝑝 ∨ 𝑞 ≡∼ 𝑝 ∧∼ 𝑞 ∼ 𝑝 ∧ 𝑞 ≡∼ 𝑝 ∨∼ 𝑞
- 5. LAWS OF ALGEBRA OF PREPOSITIONS Mayengo, M. M 5 The Law of Equivancy of Contrapositive 𝑝 → 𝑞 ≡∼ 𝑞 →∼ 𝑝 The Law of Syllogism If 𝑝 → 𝑞 and 𝑞 → 𝑟 then 𝑝 → 𝑟 Other Laws 𝑝 → 𝑞 ≡∼ 𝑝 ∨ 𝑞 𝑝 ↔ 𝑞 ≡ 𝑝 → 𝑞 ∧ (𝑞 → 𝑝)
- 6. TAUTOLOGIES AND CONTRADICTIONS Mayengo, M. M 6 Tautology A compound statement that is always true no matter what truth values are assigned to its component propositions is called a tautology Examples: Consider the following truth tables 𝒑 ∼ 𝒑 (𝒑 ∨∼ 𝒑) 𝑇 𝐹 𝑻 𝐹 𝑇 𝑻
- 7. TAUTOLOGIES AND CONTRADICTIONS Mayengo, M. M 7 Tautology Examples: Consider the following truth tables 𝒑 𝒒 ∼ 𝒑 𝒑 ∨ 𝒒 ∼ 𝒑 ∨ (𝒑 ∨ 𝒒) 𝑇 𝑇 𝐹 𝑇 𝑻 𝑇 𝐹 𝐹 𝑇 𝑻 𝐹 𝑇 𝑇 𝑇 𝑻 𝐹 𝐹 𝑇 𝐹 𝑻
- 8. TAUTOLOGIES AND CONTRADICTIONS Mayengo, M. M 8 Contradictions/Fallacy These are compound statements which are always false no matter what truth values are assigned to the component propositions Examples: Consider the following truth tables 𝒑 ∼ 𝒑 (𝒑 ∧∼ 𝒑) 𝑇 𝐹 𝑭 𝐹 𝑇 𝑭
- 9. TAUTOLOGIES AND CONTRADICTIONS Mayengo, M. M 9 Contradictions Examples: Consider the following truth tables 𝒑 𝒒 𝒑 ∧ 𝒒 𝒑 ∨ 𝒒 ∼ (𝒑 ∨ 𝒒) 𝒑 ∧ 𝒒 ∧∼ (𝒑 ∨ 𝒒) 𝑇 𝑇 𝑇 𝑇 𝐹 𝑭 𝑇 𝐹 𝐹 𝑇 𝐹 𝑭 𝐹 𝑇 𝐹 𝑇 𝐹 𝑭 𝐹 𝐹 𝐹 𝐹 𝑇 𝑭
- 10. TAUTOLOGIES AND CONTRADICTIONS Contingency These are compound statements which are neither false nor true no matter what truth values are assigned to the component propositions Mayengo, M. M
- 11. TAUTOLOGIES AND CONTRADICTIONS Mayengo, M. M 11 Logical Equivalence Two or more propositions are said to be logically equivalent (or equal) if they have the same/identical truth values. Logical equivalence is denoted by symbol " = "or " ≡ “ Examples: Consider the truth values of the propositions ∼ (𝑝 ∧ 𝑞) and ∼ 𝑝 ∨∼ 𝑞 bellow. 𝒑 𝒒 ∼ 𝒑 ∼ 𝒒 𝒑 ∧ 𝒒 ∼ (𝒑 ∧ 𝒒) ∼ 𝒑 ∨∼ 𝒒 𝑇 𝑇 𝐹 𝐹 𝑇 𝑭 𝑭 𝑇 𝐹 𝐹 𝑇 𝐹 𝑻 𝑻 𝐹 𝑇 𝑇 𝐹 𝐹 𝑻 𝑻 𝐹 𝐹 𝑇 𝑇 𝐹 𝑻 𝑻
- 12. TAUTOLOGIES AND CONTRADICTIONS Mayengo, M. M 12 Logical Equivalence The propositions ∼ (𝑝 ∧ 𝑞) and ∼ 𝑝 ∨∼ 𝑞 bellow have the same truth values for all possible ways of assigning truth values to the component propositions 𝑝 and 𝑞. Hence, we conclude that ∼ 𝑝 ∧ 𝑞 =∼ 𝑝 ∨∼ 𝑞 𝒑 𝒒 ∼ 𝒑 ∼ 𝒒 𝒑 ∧ 𝒒 ∼ (𝒑 ∧ 𝒒) ∼ 𝒑 ∨∼ 𝒒 𝑇 𝑇 𝐹 𝐹 𝑇 𝑭 𝑭 𝑇 𝐹 𝐹 𝑇 𝐹 𝑻 𝑻 𝐹 𝑇 𝑇 𝐹 𝐹 𝑻 𝑻 𝐹 𝐹 𝑇 𝑇 𝐹 𝑻 𝑻
- 13. TAUTOLOGIES AND CONTRADICTIONS Mayengo, M. M 13 Logical Equivalence Similarly, we can show that 𝑝 → 𝑞 =∼ 𝑝 ∨ 𝑞. The truth table bellow proves the equivalence of the two compound statements. 𝒑 𝒒 ∼ 𝒑 𝒑 → 𝒒 ∼ 𝒑 ∨ 𝒒 𝑇 𝑇 𝐹 𝑻 𝑻 𝑇 𝐹 𝐹 𝑭 𝑭 𝐹 𝑇 𝑇 𝑻 𝑻 𝐹 𝐹 𝑇 𝑻 𝑻
- 14. QUESTIONS By using truth table and laws of proposition logic, classify the following mathematic logic formula whether they are tautology, contradiction or contingency statements. (i)(∼p→(p→q) (ii) (p→q)(q→p) (iii) p(p→q) (iv) (pq)→(rp) (v) (pq)(∼pr) Mayengo, M. M